Quantum complexity has attracted a lot of interest across various subfields of physics, from quantum computing to the theory of black holes. In recent years, physics has offered some conjecture to bridge the gap between quantum physics and the theory of gravity and describe the behavior of complex quantum many-body systems, such as black holes and wormholes in the universe.

A theory group at Freie UniversitÃ¤t Berlin and HZB, together with Harvard University, USA, has proven a mathematical conjecture about the behavior of complexity in such systems, increasing the viability of this bridge. This study offers a strong foundation for understanding the physical properties of chaotic quantum systems, from black holes to complex many-body systems.

The evolution of generic quantum systems can be modeled by considering a collection of qubits subjected to sequences of random unitary gates. But, how many elementary operations are needed to prepare the desired state?

On the surface, it seems that this minimum number of operations – the system’s complexity – is always growing. Physicists, in this study, formulated this intuition as a mathematical conjecture: the quantum complexity of a many-particle system should first grow linearly for astronomically long times and then – for even longer – remain in a state of maximum complexity.

The behavior of theoretical wormholes motivated this conjecture. The volume of wormholes seems to grow linearly for an eternally long time. Scientists further theorized that complexity and the volume of wormholes are the same quantity from two different perspectives.

Jonas Haferkamp, a Ph.D. student in the team of Eisert and the first author of the paper, said, *“Our proof is a surprising combination of methods from geometry and those from quantum information theory. This new approach makes it possible to solve the conjecture for the vast majority of systems without tackling the notoriously difficult problem for individual states.”*

**Journal Reference:**

- Haferkamp, J., Faist, P., Kothakonda, N.B.T. et al. Linear growth of quantum circuit complexity. Nat. Phys. (2022). DOI: 10.1038/s41567-022-01539-6